Over the given field F, write f(x)=q(x)(x-c)+f(c) for (a)...

Over the given field $F$, write $f(x)=q(x)(x-c)+f(c)$ for (a) $f(x)=2 x^{3}+x^{2}-4 x+3 ; \quad c=1 ; \quad F=\mathbf{Q}$ (b) $f(x)=x^{3}-5 x^{2}+6 x+5 ; \quad c=2 ; \quad F=\mathbf{Q}$ (c) $f(x)=x^{3}+1 ; \quad c=1 ; \quad F=\mathbf{Z}_{3}$ (d) $f(x)=x^{3}+2 x+3 ; \quad c=2 ; \quad F=\mathbf{Z}_{5}$.

Over the given field $F$, write $f(x)=q(x)(x-c)+f(c)$ for
(a) $f(x)=2 x^{3}+x^{2}-4 x+3 ; \quad c=1 ; \quad F=\mathbf{Q}$
(b) $f(x)=x^{3}-5 x^{2}+6 x+5 ; \quad c=2 ; \quad F=\mathbf{Q}$
(c) $f(x)=x^{3}+1 ; \quad c=1 ; \quad F=\mathbf{Z}_{3}$
(d) $f(x)=x^{3}+2 x+3 ; \quad c=2 ; \quad F=\mathbf{Z}_{5}$.

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Key Concepts

Remainder Theorem

The Remainder Theorem is a specific case of the polynomial division algorithm when the divisor is of the form (x - c). It states that if a polynomial f(x) is divided by (x - c), then the remainder is equal to f(c). This theorem simplifies the process of finding remainders and illustrates that f(x) can be expressed as f(x) = q(x)(x - c) + f(c).

Field Arithmetic

Working over a field F, such as the rationals Q or a finite field Zn for prime n, ensures that division and other arithmetic operations behave consistently and are well-defined. This is essential when applying the polynomial division algorithm and the Remainder Theorem because the existence and uniqueness of the quotient and remainder depend on the underlying algebraic structure having properties such as the existence of multiplicative inverses for non-zero elements.

Polynomial Division Algorithm

The polynomial division algorithm states that for any polynomial f(x) and any non-zero polynomial d(x) (with d(x) typically of lower degree), there exist unique polynomials q(x) (the quotient) and r(x) (the remainder) such that f(x) = q(x)d(x) + r(x), with the degree of r(x) strictly less than the degree of d(x). In the context of dividing by a linear term (x - c), this theorem guarantees that the remainder is simply a constant.

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